{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "FD3T25A1DX"
    }
  ],
  "body": "\n## Summary\n\nIn commutative algebra, a property of [[modules|0NZ8JA25WK]] or rings is local when it holds globally exactly when it holds after [[localization at every prime ideal|X6JD4AG973]].\nLocal properties let one prove a statement by checking it over [[local rings|MZ02BKPHJA]], where tools such as [[Nakayama's lemma|15JHBYR6KH]] are available.\nBeing zero, being [[flat|73RYYEVDKA]], and the exactness of a sequence are local; being Noetherian or being an [[integral domain|0QVD7WBFGN]] is not.\n\n## Explanations\n\n### L1 human\n\nA property is local if you can test it one point at a time.\nFor a ring $R$, the \"points\" are its [[prime ideals|00MHHZ9SNE]], and testing at a point $\\mathfrak p$ means passing to the localization $M_{\\mathfrak p}$.\nFor example, a module is zero exactly when all its localizations are zero.\nSo to show that a module vanishes, it is enough to show it near every point.\nThe [[definition|FD3T25A1DX]] states the condition.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "13B30",
      "13C99"
    ],
    "name": "Local property",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Local_property"
      }
    ],
    "tag": "3S58E70Z7Z"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/3S58E70Z7Z/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/3S58E70Z7Z.md",
  "msc": [
    "13B30",
    "13C99"
  ],
  "name": "Local property",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3S5/3S58E70Z7Z.md",
  "status": null,
  "tag": "3S58E70Z7Z",
  "uri": "m2://3S58E70Z7Z",
  "url": "https://math2.tamnd.com/wiki/local-property/",
  "writing": {
    "tag": "3S58E70Z7Z",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}